Forcing axioms and the complexity of non-stationary ideals

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Introduction
The fact that closed unbounded subsets generate a proper normal filter, the club filter on κ Club κ = {A ⊆ κ | ∃C ⊆ A closed and unbounded in κ}, is one of the most important combinatorial properties of uncountable regular cardinals κ. The study of the structural properties of these filters and their dual ideals, the non-stationary ideal on κ N S κ = {A ⊆ κ | ∃Cclosed and unbounded in κ with A ∩ C = ∅} plays a central role in modern set theory.
In [23] and [24], Mekler, Shelah and Väänänen initiated the study of the complexity of club filters and non-stationary ideals, leading to various results establishing interesting connections between the complexity of these objects and their structural properties. Given an uncountable regular cardinal κ, it is easy to see that both Club κ and N S κ are definable by a 1 -formula with parameter κ, i.e. there exist 1 -formulas ϕ 0 (v 0 , v 1 ) and ϕ 1 (v 0 , v 1 ) such that Club κ = {A | ϕ 0 (A, κ)} and N S κ = {A | ϕ 1 (A, κ)}. The results of [24] show that under CH, the 1 (H(ω 2 ))-definability of N S ω 1 (i.e. the assumption that N S ω 1 = {A | ψ 1 (A, z)} holds for some 1 -formula ψ(v 0 , v 1 ) and some z ∈ H(ω 2 )) is equivalent to several interesting combinatorial and model-theoretic assumptions about objects of size ω 1 . In particular, it is shown that this definability assumption is equivalent to the existence of a so-called canary tree, a tree of height and cardinality ω 1 without cofinal branches that has specific properties with respect to the ordering of such trees under order-preserving embeddings. Since the results of [23] show that the existence of a canary tree is independent of ZFC + CH, it follows that this theory is not able to determine the exact complexity of N S ω 1 .
The above results were later generalized to higher cardinals. If S is a stationary subset of an uncountable regular cardinal κ, then we let N S S = N S δ ∩ ℘ (S) denote the restriction of the non-stationary ideal on δ to S. Given infinite regular cardinals λ < κ, we set S κ λ = {α < κ | cof(α) = λ}. In addition, if m < n < ω, then we write S n m instead of S ω n ω m . Results of Hyttinen and Rautila in [13] showed that if κ is an infinite regular cardinal in a model of the GCH, then, in a cofinality-preserving forcing extension, the set N S S κ + κ is 1 (H(κ ++ ))-definable. Furthermore, in [10], S. Friedman, Wu and Zdomskyy showed that for every successor cardinal in Gödel's constructible universe L, there is a cardinality-preserving forcing extension of L in which N S κ is 1 (H(κ + ))-definable. These results can be easily used to show that the complexity of the non-stationary ideal and its restriction is not determined by ZFC (see Lemma 1.1 and the subsequent discussion below). Finally, recent work also unveiled several interesting consequences of the 1 (H(κ + ))-definability of restriction of N S κ at higher cardinals κ. In particular, this set-theoretic assumption was shown to be closely connected to model-theoretic questions dealing with Shelah's Classification Theory and the complexity of certain mathematical theories (see, for example, [8, Theorem 64]).
The above results strongly motivate the question whether canonical extensions of ZFC decide more about the complexity of non-stationary ideals, and this question turns out to be closely connected to important recent developments in set theory. In [8], S. Friedman, Hyttinnen and Kulikov showed that, in the constructible universe L, the sets of the form N S S for some stationary subset S of an uncountable regular cardinal κ are not 1 (H(κ + ))-definable. Using the notion of local club condensation (see [7]), it is possible to extend this conclusion to larger canonical inner models. In another direction, S. Friedman and Wu observed in [9] that strong saturation properties of the non-stationary ideal on ω 1 , i.e. the assumption that the poset ℘ (ω 1 )/N S ω 1 has a dense subset of cardinality ω 1 , imply the 1 (H(ω 2 ))-definability of N S ω 1 . Results of Woodin in [26,Chapter 6] show that N S ω 1 possesses these properties in certain forcing extensions of determinacy models. Finally, Schindler and his collaborators recently studied the question whether forcing axioms determine the complexity of N S ω 1 . In [19], Larson, Schindler and Wu showed that Woodin's Axiom ( * ) (see [26, Definition 5.1]) implies that N S ω 1 is not 1 (H(ω 2 ))-definable. In combination with recent results of Asperó and Schindler in [1], this shows that MM ++ , a natural strengthening of Martin's Maximum, implies that N S ω 1 is not 1 The work presented in this paper is motivated by the question whether strong forcing axioms determine the complexity of the non-stationary ideal on ω 2 and its restrictions. The following result from [21] shows that all extensions of ZFC that are preserved by forcing with <ω 2 -directed posets are compatible with the assumption that for every stationary subset S of ω 2 , the set N S S is not 1 (H(ω 3 ))-definable. In particular, the results of [4,17,18] show that this statement is compatible with all standard forcing axioms, like MM ++ . The lemma follows directly from a combination of [21,Theorem 2.1], showing that no 1 1 -definable set (see [20, Definition 1.2]) separates Club κ from N S κ in the given model of set theory, and [20, Lemma 2.4], showing that 1 1 -definability coincides with 1 (H(κ + ))-definability at all uncountable regular cardinals κ. Lemma 1.1 Let κ be an uncountable cardinal with κ <κ = κ and let G be Add(κ, κ + )generic over V. In V[G], no 1 (H(κ + ))-definable subset of ℘ (κ) separates Club κ from N S κ , i.e. no set X definable in this way satisfies Club κ ⊆ X ⊆ ℘ (κ) \ N S κ .
Note that, if S is a stationary subset of an uncountable regular cardinal κ, then N S S separates Club κ from N S κ . This shows that, in Add(κ, κ + )-generic extensions, sets of the form N S S for stationary subsets S are not 1 (H(κ + ))-definable.
In contrast, we will prove the following theorem that shows that strong forcing axioms like MM ++ are also compatible with the existence of a 1 (H(ω 3 ))-definable set that separates the club filter on ω 2 from the corresponding non-stationary ideal. The proof of this result is based on a detailed analysis of the preservation properties of a variation of a forcing iteration constructed by Hyttinen and Rautila in the consistency proofs of [13]. Our construction will also allow us to produce such models with arbitrary large 2 ω 2 . See Sect. 2 for the meaning of the "+μ" versions of forcing axioms. 1 (2) A set g is (W , P)-generic if g ⊆ P ∩ W , g is a filter on P ∩ W , and D ∩ g = ∅ for every D ∈ W that is a dense subset of P. 2 (3) A condition p ∈ P is a (W , P)-total master condition if the set {r ∈ P ∩ W | p ≤ P r } is a (W , P)-generic filter.
The following result is well-known: Lemma 2.2 Let P be a poset, let W ≺ (H(θ ), ∈, P), let μ be an ordinal with μ ⊆ W , and letḟ ∈ W be a P-name for a function from μ to the ground model V. If p is a (W , P)-total master condition and G is P-generic over V with p ∈ G, thenḟ G ∈ V.
Proof Fix a (W , P)-total master condition p and a filter G on P that is generic over V and contains the condition p. Let g = {r ∈ P ∩ W | p ≤ P r } denote the (W , P)generic filter induced by p. Then G ∩ W is a (W , P)-generic filter extending g and therefore standard arguments show that G ∩ W = g. By elementarity, there is a sequence A ξ | ξ < μ ∈ W of maximal antichains in P with the property that for every ξ < μ, each condition in A ξ decides the value ofḟ at ξ . Since μ ⊆ W , this shows that for all ξ < μ, the unique condition in A ξ ∩ g decides the value ofḟ at ξ . But the sequence A ξ | ξ < μ and the filter g are both elements of V and hence the functionḟ G is also in the ground model.
We state a definition that will be used extensively in the following arguments: Definition 2.3 Given an infinite regular cardinal κ, we let IA κ denote the class of all sets W with the property that there exists a sequence N = N α | α < κ that satisfies the following statements: (1) The sequence N is ⊆-increasing and ⊆-continuous.
(4) Every proper initial segment of N is an element of W .

Remark 2.4
If N witnesses that W is an element of IA κ and W ≺ H(θ ) for some θ > κ, then κ ⊆ W . This is because we have N α ∈ W for every α < κ, and the domain of N α, namely α, is definable from the parameter N α.
In what follows, if τ is a regular uncountable cardinal, ℘ τ (H ) refers to the set of all W ⊆ H with |W | < τ, and ℘ * τ (H ) denotes the set The set ℘ * τ (H(θ )) contains a club in the sense of Jech (see [14]), but not necessarily in the sense of Shelah (see [6]).

Lemma 2.6
If κ is a regular and uncountable cardinal, then IA κ is stationary in ℘ κ + (H(θ )) for all sufficiently large regular θ .
Proof Given a first-order structure A = (H(θ ), ∈, κ, . . .) in a countable language, recursively construct a ⊆-continuous and ⊆-increasing sequence N = N α | α < κ of elementary substructures of A of cardinality less than κ such that N α ∈ N α+1 for all α < κ. Then N witnesses that its union is contained in IA κ . Lemma 2.7 Let P be a poset, let κ < θ be infinite regular cardinals with P ∈ H(θ ), let be a well-ordering of H(θ ), let W ≺ (H(θ ), ∈, P, ) with W ∈ IA κ , and let p ∈ P ∩ W .
(1) If P is <κ-closed, then there exists a (W , P)-generic filter that contains p. (2) If P is <κ + -closed, then there exists a (W , P)-total master condition below p.
(1) Assuming that P is <κ-closed. Using the closure of P and the fact that each N α has cardinality less than κ, we can recursively construct a descending sequence p = p α | α < κ of conditions below p in P such that the following statements hold for all α < κ: (a) The condition p α+1 is the -least element of P below p α that is an element of every open dense set that belongs to N α . 4 (b) If α is a limit ordinal, then p α is the -least lower bound of the sequence p | < α .
Then every proper initial segment of p is definable from a proper initial segment of N , and hence every proper initial segment of p is in W . In particular, we know that p α+1 ∈ W for all α < κ. It follows that the filter in P generated by the subset { p α | α < κ} is (W , P)-generic.
(2) Now, assume that P is <κ + -closed and repeat the above construction of the sequence p. Then p has a lower bound in P, and this lower bound is clearly a (W , P)total master condition.
Next we discuss one variant of proper forcing. (1) A poset P is I A κ -proper if for all sufficiently large regular cardinals θ , all W ≺ (H(θ ), ∈, P) with W ∈ IA κ and all p ∈ P∩ W , there is a (W , P)-master condition below p. (2) A poset P is I A κ -totally proper if for all sufficiently large regular cardinals θ , all W ≺ (H(θ ), ∈, P) with W ∈ IA κ and all p ∈ P ∩ W , there is a (W , P)-total master condition below p.
It is well-known that IA κ -proper posets preserve all stationary subsets of S κ + κ that lie in the approachability ideal I [κ + ] defined below. Since we could not find a reference for exactly what is needed in our arguments, we sketch the proof below. Note that it is possible for IA κ -proper (even IA κ -totally proper) posets to destroy the stationarity of some subsets of S κ + κ (see [5]). So IA κ -total properness is, in general, strictly weaker than κ + -Jensen completeness (defined in the next section), because <κ + -closed forcings preserve all stationary subsets of κ + . Definition 2.9 (Shelah) Let κ be an infinite regular cardinal.
(1) Given a sequence z = z α | α < κ + a sequence of elements of [κ + ] <κ , an ordinal γ < κ + is called approachable with respect to z if there exists a sequence cofinal in γ such that every proper initial segment of α is equal to z α for some α < γ . (2) The Approachability ideal I[κ + ] on κ + is the (possibly non-proper) normal ideal generated by sets of the form Note that a subset X of κ + is an element of I [κ + ] if and only if there exists some club D ⊆ κ + and some sequence z ∈ κ + ([κ + ] <κ ) such that every γ ∈ D ∩ X is approachable with respect to z. In the following, we will make use of several facts about I [κ + ]. Throughout this section, κ denotes a regular cardinal. Lemma 2.10 ( [5]) Suppose κ <κ ≤ κ + , and let z α | α < κ + be an enumeration of [κ + ] <κ . 5 Define Then the following statements hold: Proof (1) Fix a sufficiently large regular cardinal θ and a well-ordering of H(θ ). Fix W ∈ IA κ with W ≺ (H(θ ), ∈, , z) and let N α | α < κ be a sequence witnessing that W ∈ IA κ . Given α < κ, set γ α = sup(N α ∩ κ + ) < κ + . Then γ = γ α | α < κ enumerates a cofinal subset of W ∩κ + of order-type κ and every proper initial segment of this sequence is an element of W . Moreover, each proper initial segment of γ is an element of [κ + ] <κ , and hence an element of This shows that W ∩κ + is approachable with respect to z. Since Lemma 2.6 shows that there are stationarily-many W ∈ IA κ with W ≺ (H(θ ), ∈, , z), these computations allow us to conclude that M z is a stationary subset of S κ + κ .
(3) Now, suppose that S ∈ I [κ + ] is a stationary subset of S κ + κ . By earlier remarks, there is a sequence u = u α | α < κ + of elements of [κ + ] <κ and club subset D of κ + with the property that every γ ∈ D ∩ S is approachable with respect to u. Define In particular, every proper initial segment of β appears in z before γ and therefore γ is approachable with respect to z. These computations show that S \ M z ⊆ S \ E is non-stationary in κ + .
The next few lemmas address stationary set preservation when GCH may fail to hold.

Lemma 2.11
The class IA κ is projective stationary over Then there is a club D in κ + and a sequence z ∈ κ + [κ + ] <κ such that every element of D ∩ T is approachable with respect to z. Fix a regular ϑ with F ∈ H(ϑ), let be a well-ordering of H(ϑ) and set Pick γ ∈ D ∩ T and W ≺ A with γ = W ∩ κ + , which is possible because T is stationary. Since γ ∈ T ∩ D, there is an increasing sequence β = β α | α < κ that is cofinal in γ and has the property that every proper initial segment of β is equal to z α for some α < γ . Since z ∈ W and W ∩ κ + = γ , it follows that every proper initial segment of β is an element of W . Recursively define a sequence N = N α | α < κ as follows: On the other hand, for each α < κ, the sequence N | ≤ α is definable in A from the parameter β | ≤ α , which is an element of W by the above remarks. Hence every proper initial segment of N is an element of W and, in particular, we know that It follows that sup(N ∩κ + ) ≤ γ . Combined with (1), this shows that sup(N ∩κ + ) = γ . Finally, since α ⊆ N α+1 for all α < κ, it follows that κ ⊆ N and hence we know that N ∩ κ + is transitive. This allows us to conclude that completing the proof of the lemma.
The following lemma is one way to salvage stationary set preservation in the non-GCH context.

Lemma 2.12
Let P be a IA κ -proper poset and let T ⊆ S κ + κ be stationary with T ∈ I [κ + ]. Then forcing with P preserves the stationarity of T .
Proof Set τ = κ + . LetĊ be a P-name for a club in τ , let p ∈ P and let θ be a sufficiently large regular cardinal. Using Lemma 2.11, we find γ ∈ T and W ≺ (H(θ ), ∈, p,Ċ, P) with W ∈ IA κ and W ∩τ = γ . By our assumptions, there is a (W , P)-master condition q below p in P. Let G be P-generic over V with q ∈ G. Then W [G]∩τ = W ∩τ = γ . Moreover, sinceĊ ∈ W , we now know thatĊ G ∩ W [G] is unbounded in γ and hence γ ∈Ċ G ∩ T .
These computations show that, in the ground model V, we have for densely-many conditions q in P.

Generalizing a lemma of Jensen
The notion of IA κ -properness, defined in Sect. 2, is a non-GCH analogue of the notion of κ-properness introduced in [13, Definition 3.4] . This notion will be important to proving that tails of the iteration described in Sect. 4 do not add cofinal branches to a certain tree, and that argument will closely follow the corresponding arguments of [13]. However, IA κ -properness (in the case κ = ω 1 ) is not sufficient for ensuring the preservation of forcing axioms that we need for the proofs of our main results. There are examples of IA ω 1 -proper forcings that destroy, for example, the Proper Forcing Axiom. 6 On the other hand, <ω 2 -directed closed posets preserve all standard forcing axioms (see [17] and [18]). In this section, we generalize a result of Jensen, yielding a property that is forcing equivalent to <ω 2 -directed closure, but often easier to verify than <ω 2 -directed closure.
In [16], Jensen defines a poset P to be complete if for every sufficiently large θ , there are club-many W ∈ ℘ ω 1 (H(θ )) such that every (W , P)-generic filter has a lower bound in P. 7 He then proves: (1) The poset P is complete.
(2) The poset P is forcing equivalent to a σ -closed poset.
We will generalize a version of this lemma to larger cardinals, and, in fact, characterize directed closure (see Lemma 3.6 below). 8 However there are a few technicalities to address. Note that for any W ∈ ℘ ω 1 (H(θ )), the fact that W is countable ensures that there always exist (W , P)-generic filters, regardless of what P is. In particular, the phrase " . . . every (W , P)-generic filter … " is never vacuous, if W is countable. Of course, for uncountable W , it may happen that (depending on the poset P) there do not exist any (W , P)-generic filters at all; e.g. if W ≺ H(θ ) and ω 1 ⊆ W , then there does not exist a (W , Col(ω, ω 1 ))-generic filter. Definition 3.2 Given a regular uncountable cardinal τ , a poset P is τ -Jensen-complete if the following statements hold for all sufficiently large regular cardinals θ : (1) For every p ∈ P, there are stationarily-many W ∈ ℘ * τ (H(θ )) with the property that there exists a (W , P)-generic filter including p.
(2) For all but non-stationarily many W ∈ ℘ * τ (H(θ )), every (W , P)-generic filter has a lower bound in P. 9 Remark 3.3 Note that clause (1) of Definition 3.2 always holds true for τ = ω 1 , and is hence redundant in that case. In particular, for τ = ω 1 , Definition 3.2 is equivalent to Jensen's definition of completeness.

Remark 3.4
In combination, the clauses (1) and (2) of Definition 3.2 imply that the poset P is totally proper on a stationary subset of ℘ * τ (H(θ )); i.e. that there are stationarily-many W ∈ ℘ * τ (H(θ )) such that every condition in P ∩ W can be extended to a (W , P)-total master condition in the sense of Definition 2.1. This conclusion, however, is strictly weaker than τ -Jensen-completeness, since (for example with τ = ω 1 ) shooting a club through a bistationary subset of ω 1 has the latter property but is not <ω 1 -closed. In the case τ = ω 2 , if 2 ω 1 = ω 2 , then shooting an ω 1 -club through the set M described in Lemma 2.10 is IA ω 1 -totally proper, but forces the approachability property to hold at ω 2 . In particular, this forcing destroys the Proper Forcing Axiom, and PFA is preserved by ω 2 -Jensen-complete forcings (by [17] and Lemma 3.6 below).

Lemma 3.5 If κ is an infinite cardinal and P is a <κ-closed poset, then clause (1) of Definition 3.2 holds for τ = κ + and P.
Proof This follows immediately from Lemmas 2.6 and 2.7.
Next, we state our generalization of Jensen's lemma. Its Corollary 2 will be used in the proof of Theorem 4.2 below. Lemma 3.6 Given a poset P and a successor cardinal τ , the following statements are equivalent: (1) The poset P is forcing equivalent to a <τ -directed closed poset.
(2) The poset P is forcing equivalent to a τ -Jensen-complete poset.
Proof First, assume that P is <τ -directed closed. Then, in particular, P is <τ -closed, and hence Lemma 3.5 ensures that clause (1) of Definition 3.2 holds for P. But then the directed closure of P ensures that any (W , P)-generic filter for any W ∈ ℘ * τ (H(θ )) has a lower bound in P, and hence clause (2) of Definition 3.2 holds for P as well. This shows that P is τ -Jensen-complete. Now, suppose that P is τ -Jensen-complete. Let F : [H(θ )] <ω −→ H(θ ) generate a club witnessing clause (2) of Definition 3.2, i.e. whenever W ∈ ℘ * τ (H(θ )) and W is closed under F, then any (W , P)-generic filter has a lower bound. We may assume that F also codes a well-ordering of H Define a poset Q, whose conditions are pairs (M, g) satisfying the following statements: and whose ordering is given by: Note that ≤ Q is transitive and clause (1) of Definition 3.2 ensures that Q is nonempty.
The regularity of τ ensures that M ∈ ℘ * τ (H(θ )), and M is closed under F, because each M i is closed under F and the collection M i | i ∈ I is ⊆-directed. In addition, we have g ⊆ M ∩ P and g clearly has the property that D ∩ g = ∅ for every dense D ∈ M, because each such D lies in some M i and g i ⊆ g is an (M i , P)-generic filter. Finally, the fact that g is a filter follows easily from the fact that the given collection is directed and each g i is a filter. This shows that (M, g) is a condition in Q. Now, fix i ∈ I . Then M ⊇ M i , g∩M i is a filter on M i ∩P, and g∩M i ⊇ g i . But since g i is (M i , P)-generic, we know that g i is a ⊆-maximal filter on M i ∩P. In particular, we can conclude that g ∩ M i = g i . This computation shows that (M, g) ≤ Q (M i , g i ).

Claim 3.8
The poset Q is forcing equivalent to P.

Proof of Claim 3.8
It is easy to see that the boolean completions of τ -Jensen-complete posets are themselves τ -Jensen-complete. Therefore, we may assume that P is a complete boolean algebra. For each condition (M, g) in Q, let p M,g be the P-greatest lower bound of g. This conditions exists and is non-zero, because M is closed under F, g is (M, P)-generic, P is a complete boolean algebra, and because of clause (2) of Definition 3.2. In the following, we will show that the map e : Q −→ P; (M, g) −→ p M,g is a dense embedding, which will finish the proof of the claim.
First, we show that e is order-preserving. Suppose that (N , h) ≤ Q (M, g). Then N ⊇ M and g = h ∩ M. Since g ⊆ h and p N ,h is a lower bound of h, it follows that p N ,h is also a lower bound of g. But p M,g is the greatest lower bound of g, and hence Next, we show that e preserves incompatibility. Suppose (M 0 , g 0 ) and (M 1 , g 1 ) are conditions in Q with the property that there is a condition p in P that extends both e(M 0 , g 0 ) and e(M 1 , g 1 ). By clause (1) Furthermore, since p is below both p M 0 ,g 0 and p M 1 ,g 1 and M i ∩ P ⊆ W ∩ P for all i < 2, the fact that g 0 and g 1 are maximal filters in M 0 ∩ P and M 1 ∩ P, respectively, implies that G ∩ M 0 = g 0 and G ∩ M 1 = g 1 .
Finally, we show that the range of e is dense in P. Fix a condition p in P. By clause This completes the proof of the lemma.

Remark 3.9
Another common way to verify the <τ -distributivity of a given poset P is the following weaker version of τ -Jensen completeness: if for every p ∈ P, there are stationarily-many W ∈ ℘ * τ (H(θ )) such that there is a (W , P)-total master condition below p (see Definition 2.1), then P is <τ -distributive. Note that this weaker version would not suffice for our purposes, however, because we seem to need <τ -directed closure (or a close approximation of it) to prove Theorem 4.2.
Corollary 2 Let κ be an infinite regular cardinal and set τ = κ + . If P is <κ-closed poset with the property that for all but non-stationarily many W ∈ ℘ * τ (H(θ )), every (W , P)-generic filter has a lower bound in P, then the poset P is forcing equivalent to a <τ -directed closed poset.
Proof By Lemma 3.5, the <κ-closure of P ensures that clause (1) of Definition 3.2 holds. Since clause (2) of Definition 3.2 holds by assumption, this implies P is τ -Jensen-complete and Lemma 3.6 yields the desired conclusion.
In particular, if a poset P satisfies the assumptions of the above corollary for κ = ω 1 , then forcing with P preserves all standard forcing axioms.

Definition 4.1 If κ be an infinite regular cardinal and let S ⊆ S κ +
κ . Then we let T (S) denote the tree that consists of all t ∈ <κ + κ + such that dom(t) is a successor ordinal, ran(t) ⊆ S, t is strictly increasing, and t is continuous at all points of cofinality κ in its domain and is ordered by end-extension.
Note that, in the situation of the above definition, the tree T (S) has height κ + and contains a cofinal branch if and only if the set S contains a κ-club.
We are now ready to state the aspired result.
, there is a subtree T of <κ + κ + of height κ + without cofinal branches such that the following statements hold:

is stationary, then there is an order-preserving function from T (S) to T .
Note that the above theorem directly generalizes the main result of [13]: if κ <κ = κ holds, then part (4) of Lemma 2.10 shows that S κ + κ is an element of I [κ + ], and hence is the tree given by the theorem, then there is an order-preserving function from the tree . In particular, this shows that T is a κ-canary tree (see [13,Definition 3 i.e. if S is a stationary subset of S κ + κ and P is a <κ + -distributive poset that forces κ + \ S to contain a club subset, then forcing with P adds a cofinal branch to T .
For the remainder of this section, fix an infinite regular cardinal κ. Until further notice, we do not make any cardinal arithmetic assumptions.
In the following, we closely follow the arguments on pages 1684-1692 of Hyttinen-Rautila in [13], which assumed GCH (in particular, their arguments heavily rely on the assumption κ <κ = κ). We also follow their notation as closely as possible.  12 and whose ordering is given by reversed inclusion.

Proposition 4.4 The poset
Proof This statement follows directly from the fact that the union f of a coherent collection of conditions in Q 0 still has the required property that f (η) f (β) for all η < β in the domain of f , and, if the union is of size less than κ + , then the domain of f has size less than κ + too.

Remark 4.6
In the situation of the above definition, the tree T (G 0 ) has height κ + , since for any β ∈ S κ + κ , the function f with domain β and constant value β + 1 has the property that for all δ ∈ S κ + κ with δ ≤ dom( f ), the restriction f δ is not a function from δ to δ and hence cannot be the same as the function ( G 0 )(δ).
Proof Work in V and assume, towards a contradiction, that a condition f in Q 0 forces a Q 0 -nameḃ to be a cofinal branch throughṪ (Ġ 0 ). Using Proposition 4.4, easy closure arguments allow us to find λ ∈ S κ + κ , a function h : λ −→ λ and a condition g below f in Q 0 such that λ = sup(dom(g)) and g forces h to be the restriction ofḃ to λ. By the definition of T (G 0 ), this implies that h δ = g(δ) holds for all δ ∈ dom(g) and we can conclude that g ∪ {(λ, h)} is a condition in Q 0 below g. But this condition forces that h is not contained inṪ (Ġ 0 ), a contradiction.
The following poset, again taken from [13], adds an order preserving function from T (S) to T (G 0 ). The role of clause 3(a)i is to add such a function with initial segments. However, the role of clauses 3(a)ii through 3(a)vi is not obvious; roughly, with the exception of clause 3(a)iv, these properties allow us to verify κ + -Jensen completeness by ensuring the existence of lower bounds for any generic filter over any κ-sized elementary submodel. The role of clause 3(a)iv is to ensure that no cofinal branch is added to T (G 0 ). ( 13 I.e. a model of ZFC in which V[G] is a transitive class. (3) We let P(S, G 0 ) denote the unique poset defined by the following clauses: (a) A condition in P(S, G 0 ) is a pair (h, X ) satisfying the following statements: (i) h is an order-preserving partial function of cardinality at most κ from the tree T (S) to the tree T (G 0 ) with the property that dom(h) is closed under initial segments. and

Remark 4.9
In the above definition, the requirements on X (α) differ depending on whether or not cof(α) = κ. If α ∈ S κ + κ ∩ dom(X ), then X (α) is a proper initial segment of the function ( G 0 )(α). 14 In combination with requirement (3(a)v) in the above definition, this shows that for all α ∈ dom(X ) ∩ S κ + κ , there is an ordinal η α < α such that no node in the range of h can extend ( G 0 )(α) η α . On the other hand, if α ∈ dom(X ) with cof(α) < κ, then the only requirement on X (α) is that nothing in the range of h is allowed to extend X (α).
As pointed out near the bottom of page 1684 of [13], the poset P(S, G 0 ) is <κclosed. Requirements (3(a)iii) and (3(a)vi) of Definition 4.8 are mainly needed for the proof of Lemma 4.10 below.
For Statement (4), let M and y ∈ M be as stated, and suppose for a contradiction there exists γ < δ with cof(γ ) Work in M. The statements (1), (2), and (3) are obviously upward absolute from V 1 [K ] to M. By Statement (2), we know that γ is in the domain of X K , and, by applying Remark 4.9 to some condition in K whose second coordinate has γ in its domain, we can find ρ γ < γ with X K (γ ) = ( G 0 )(γ ) ρ γ . Note that, by the definition of Q 0 , we know that ( G 0 )(γ ) is a total function on γ , and therefore our assumption implies that γ ≤ dom( f y ). Then ρ γ ∈ dom( f y ), and there is some ζ * < λ with ρ γ ∈ dom(y ζ * ). In particular, we know that But this implies that y ζ * ∈ ran(h K ) extends X K (γ ) ∈ ran(X K ), contradicting Statement (3).
We now describe the iteration that will witness the poset from Theorem 4.2. This is a slight variant of the iteration described at the bottom of page 1684 of [13].
The main differences are: • The length of our iteration is at least 2 2 κ . This is to allow for the case when, in the ground model, the cardinal 2 κ + is very large. • More significantly, at a given stage α of our iteration, when considering the setṠ α given to us by the bookkeeping device, we only force with the poset P(Ṡ α , G 0 ) if the statement " S κ + κ \Ṡ α contains a stationar y set in I [κ + ] " holds in the corresponding generic extension of the ground model. This will ensure (via an application of Lemma 2.12) that the complements of theṠ α 's remain stationary throughout the iteration, which in turn will be the key to showing that the tree T (G 0 ) has no cofinal branch in the final model.

Remark 4.11
In the GCH setting of [13], requiring (2) to hold is no restriction at all, since in that scenario, this statement holds for every set bistationary in S κ + κ . But, if κ <κ > κ holds, then the requirement (2) seems to be needed in order to prove that the iteration adds no cofinal branch to the tree T (G 0 ).
In the following, we fix a cardinal ε satisfying ε 2 κ = ε. Let C denote the set of all partial functions from ε to H(κ + ) of cardinality at most κ.
Then our assumptions on ε imply that |C| = ε. Next, let N denote the set of all partial functions from S κ + κ × 2 κ to C. Again, our assumptions imply that |N | = ε and we can pick an ε-to-one surjection b : ε −→ N .

Definition 4.12 We define
to be a <κ + -support iteration satisfying the following clauses: (1)Q 0 is chosen in a canonical way that ensures that the map i : Q 0 −→ P 1 ; q −→ q is an isomorphism.
If we defineḂ then there exists a P α -nameṠ α for a subset of S κ + κ such that the following statements hold in V[G] whenever G is P α -generic over V and G 0 is the induced Q 0 -generic filter over V: (3) Assume that α ∈ [1, ε) has the property that the poset P α is <κ + -distributive and there exists no sequence of conditions in P α with the properties listed in (2). Theṅ whenever G is P α -generic over V and G 0 is the induced Q 0 -generic filter over V. (4) If α ∈ [1, ε) has the property that the poset P α is not <κ + -distributive, thenQ α is a P α -name for a trivial poset.
Remark 4. 13 We include the cases (2c) and (3) in the above definition of the nameṠ α to simplify notation later on. Note that, since we have T (∅) = ∅, conditions in P(∅, G 0 ) always have trivial first coordinate, and the poset P(∅, G 0 ) is forcing equivalent to Add(κ + , 1).
Throughout the rest of this paper, P refers to the poset P ε . Moreover, in order to conform to the notation from [13], if G is P α -generic over V for some α ≤ ε, then we let G 0 denote the induced Q 0 -generic filter over V.
Definition 4.14 A condition p in P is called flat if there exists a sequence x α | i ∈ sprt( p) with the property that x α ∈ H(κ + ) and hold for all α ∈ sprt( p) with the property that P α is <κ + -distributive.
Just as in [13], the flat conditions turn out to be dense in P, as we will see in Lemma 4.20 below.
Although the density of the flat conditions is not needed to prove the κ + -Jensencompleteness of P in Lemma 4.20, it will be crucial for the proofs of the following statements: • The " tails " of the above iteration are proper with respect to IA κ (see Lemma 4.27), which in turn is important for the proof that the tree T (G 0 ) has no cofinal branches in P-generic extensions of V (see Lemma 4.28). • If 2 κ = κ + , then P satisfies the κ ++ -chain condition.
The function p( f 0 , g) defined in Definition 4.15 below is a natural attempt to form a flat condition out of a (W , P)-generic filter for some elementary substructure W of size κ. W ≺ (H(θ ), ∈, P) with |W | = κ ⊆ W , and g ⊆ P ∩ W is a (W , P)-generic filter in V.
Note that, in the last part of the above definition, the function p( f 0 , g) may or may not be a condition in P. The following lemma shows how we can ensure that p( f 0 , g) is a flat condition below every condition in g. (1) p( f 0 , g) is a flat condition in P that extends every element of g. (2) There is an α ∈ W ∩ [1, ε) such that the following statements hold: (a) p( f 0 , g) α is a condition in P α that extends every element of g α . (b) If τ ∈ W is a P α -name for a function from κ to the ordinals, then (c) There is a condition q in P α below p( f 0 , g) α with the property that the following statements hold true in V[G], whenever G is P α -generic over V with q ∈ G: Proof Set g ε = g and, given β ≤ ε, let β denote the statement asserting that p( f 0 , g) β is a flat condition in P β that lies below every element of g β . Suppose that ε fails, i.e. that part (1) of the disjunctive conclusion of the statement of the lemma fails. Let β ≤ ε be the least ordinal such that β fails.

Claim 4.17 β is a successor ordinal and an element of W .
Proof of Claim 4.17 First, we have β > 0, because the 0th component of p( f 0 , g) is f 0 , which is assumed to be a condition stronger than g 0 , and g 0 is a (W , Q 0 )-generic filter. Now, assume, towards a contradiction, that β is a limit ordinal. Since α holds for all α < β and since the support of p( f 0 , g) is contained in the κ-sized set W ∩ ε, it follows easily that p( f 0 , g) β is a condition and is below every element of g β .
Furthermore, for each α ∈ W ∩ β, let Then for all α 0 < α 1 < β, it follows easily that So the x α 's are coherent, and their union witnesses flatness of p( f 0 , g) β. This shows that β holds, a contradiction.
The above computations yield an ordinal α with β = α + 1. Assume, towards a contradiction, that α / ∈ W . Note that, if r ∈ g, then r ∈ W and, since sprt(r ) is a κ-sized element of W and κ ⊆ W , it follows that sprt(r ) ⊆ W .
Since α is not an element of W , this shows that g α = g β and p( f 0 , g) β = p f 0 , g) α. But, since α holds, this immediately implies that β holds too, a contradiction.
The above claim shows that there is an α ∈ W ∩ [1, ε) with β = α + 1. We claim that this α witnesses part (2) of the conclusion of the lemma holds true. By the minimality of β, we know that (2a) holds and P α is <κ + -distributive. Moreover, since p( f 0 , g) α is a condition that extends the (W , P α )-generic filter g α , part (2b) holds by Lemma 2.2.

Claim 4.18 There is an x
Furthermore, if G is P α -generic over V with p( f 0 , g) α ∈ G, then the following statements hold in V[G]: (1) The pair (ḣ G g,α ,Ẋ G g,α ) satisfies all requirements to be a condition in the poset P(Ṡ G α , G 0 ), with the possible exception of requirement (3(a)vi) of Definition 4.8. In particular, the following statements hold:

(b) Every proper initial segment of ( G 0 )(W ∩ κ + ) is an element of W , and is anṠ G α -node. (c) No proper initial segment of ( G 0 )(W ∩ κ + ) is an element of ran(Ẋ G g,α ).
Proof of Claim 4. 18 In order to make use of Lemma 4.10, it will be more convenient to work with the transitive collapse of W instead of W itself. Let H W be the transitive collapse of W , and let σ : H W −→ W ≺ H(θ ) be the inverse of the collapsing map. In the following, if b is a set, then we will writē Note thatb = σ −1 (b) holds for all b ∈ W and we will frequently use this abbreviation in the following arguments.
Since g is a (W , P)-generic filter, we know thatḡ is aP-generic over H W , and, in particular, it follows thatḡ γ isP γ -generic over H W for all γ ∈ W ∩ ε.
If we define Since α holds, we know that p( f 0 , g) α is a condition in P α that extends every element of the (W , P α )-generic filter g α . In particular, p( f 0 , g) α it is a total master condition for W . By Lemma 2.2, every P α -name for a function from κ to the ordinals in W is forced by p( f 0 , g) α to be evaluated to an element of W . It follows that Let h k be the union of the left coordinates of k and let X k be the union of the right coordinates of k.
By (4), we can apply Lemma 4.10 with the ground model H W and the outer model H W [ḡ α ] and derive the following statements: • The function is order preserving and every element of its range is a σ −1 (Ṡ α )ḡ α -node in H W [ḡ α ].
• No element of ran(h k ) extends an element of the range of the function Set x = (h k , X k ). In the following, we will show that p( f 0 , g) α and x satisfy (3), and p( f 0 , g) α forces the other statements of the claim to hold true.
By the definition ofσ , we now know that This shows thatċ G g,α ⊆σ [k] and, together with the above computations, we can conclude thatσ By (5)  Since the critical point ofσ is δ, it follows that k, h k and X k are all pointwise fixed byσ . In particular, we have Since p( f 0 , g)(α) is, by definition, the P α -name pair P α (ḣ g,α ,Ẋ g,α ), this completes the proof of (3). Part (1) of the claim follows by the properties of (h k , X k ) over H W [ḡ α ] discussed above, together with the equality (6), elementarity ofσ , and the fact thatσ fixes bounded subsets of δ that lie in H W [ḡ α ]. For example, to verify requirement (4.8(a)iv) of Definition 4.8, suppose t is in the range of h k . Then t is in the range of the left coordinate of some condition in k ⊆σ −1 (P(Ṡ G α , G 0 ) V[G] ), and hence t is anσ −1 (Ṡ G α )node in H W [ḡ α ]. By elementarity ofσ and the fact thatσ fixes y, it follows that t is The remaining requirements of Definition 4.8, except for requirement (3(a)vi), are easily verified for the pair displayed in (6) in a similar manner. Now, we prove that p( f 0 , g) α forces the statements in part (2) of the claim. Recall G is an arbitrary P α -generic filter over V with p( f 0 , g) α ∈ G. Assume that the ordered pair (6) is not a condition in P In the following, we show that the statements (2a), (2b), and (2c) of the claim hold true in V[G]. By part (1) of the claim, it must be requirement (3(a)vi) of Definition 4.8 that fails. In particular, there is an increasing sequence y = y ζ | ζ < κ of nodes in the range of h k in V[G] such that the function fȳ = ζ <κ y ζ is not an element of T (G 0 ). Since the elements of the range of h k are κ-sized objects in H W [ḡ α ] and hence in H W by (4), this implies that the domain of f y is at most δ = (κ + ) H W . In summary, there is some ordinal γ ∈ (S κ + κ ) V such that By part (4) In summary, we have shown that W ∩ κ + = dom( f y ) and Since f y is a union of functions in the range of h k , and (4) together with the fact that the critical point of σ is δ imply that h k ⊆ W , every proper initial segment of f y is an element of W . Furthermore, every proper initial segment of f y is extended by some y ∈ ran(h k ), which is anṠ G α -node in V[G] by part (1) of the claim. Hence, every proper initial segment of f y is anṠ G α -node in V[G], and is an element of W . Together with (8), this proves part (2b) of the claim. Finally, to prove part (2c) of the claim, suppose γ ∈ dom(X k ), define η = dom(X k (γ )) and assume, towards a contradiction, that f y η =X (α). Note that η < δ, because X k ⊆ H W [ḡ α ]. In particular, we have y ζ η = X k (α) for some ζ < κ. But this contradicts the fact from part (1) that nothing in the range of h k extends any function from the range of X k .
It remains to prove part (2c) of the lemma, which will essentially follow from part (2) of Claim 4.18, though we first must dispense with a technicality. Recall that α+1 fails, but α holds. Next, we observe that the failure of α+1 is due to the function p( f 0 , g) (α + 1) not being a condition at all (rather than being a condition but failing to extend g α+1 , or being a condition but failing to be flat):

Claim 4.19 Some condition in
is not a condition inQ α .

Proof of Claim 4.19
Assume not, i.e. suppose that p( f 0 , g) α forces that the pair in (9) to be a condition inQ α . Since the components of the pair in (9) are given by the union of the left and right coordinates ofċ g,α , the fact that the ordering ofQ α is given reversed inclusion now implies that the condition p( f 0 , g) α forces p( f 0 , g)(α) to be stronger than every condition inċ g,α . Since the validity of α implies that p( f 0 , g) α is stronger than every condition in g α , it follows that is stronger than every condition in g α+1 . Furthermore, by Claim 4.18, there is an x α ∈ V such that Since α holds, we know that p( f 0 , g) α is flat. Let x | ∈ W ∩ α witness its flatness. Then the sequence x | ∈ W ∩ (α + 1) witnesses the flatness of p( f 0 , g) (α + 1). In summary, p( f 0 , g) (α + 1) is a flat condition below every member of g α+1 , contradicting the fact that α+1 fails. Part (2c) of the lemma now follows immediately from Claim 4.19, and part (2) of Claim 4.18.
The above results now allow us to prove the following key lemma.

Lemma 4.20
The poset P is κ + -Jensen complete, and the flat conditions are dense in P.
Before we prove this result, we make a couple of remarks.

Remark 4.21
In [13, Claim 3.13], a weaker version of Lemma 4.20, stating that P is κ-proper, was proven. This concept was defined in [13,Definition 3.4] and only makes sense under the assumption that κ <κ = κ. It, in particular, implies that the given poset is <κ + -distributive.
In the non-GCH setting, in particular, when we do not assume κ <κ = κ, perhaps the most natural analogue of κ-properness is our notion of IA κ -proper (Defininition 2.8). In fact, changing just a few words in the proof of [13,Claim 3.13] would suffice to prove that (even without GCH) the poset P is proper for IA κ and is <κ + -distributive. However, that conclusion does not suffice for applications in our main theorems, since, for example, IA ω 1 -properness, even together with <ω 2 -strategic closure, does not guarantee preservation of the Proper Forcing Axiom. 18 We seem to need the stronger property of κ + -Jensen completeness (i.e. <κ +directed closure), which we prove in Lemma 4.20. This requires some reorganization and strengthening of the argument of [13,Claim 3.13], but the main ideas of the proof of Lemma 4.20 are very similar to the proof of [13,Claim 3.13].

Remark 4.22
Iterations using <κ + -support, where each iterand is <κ + -directed closed, are themselves <κ + -directed closed. However, this fact seems to not be applicable to the iteration P ε constructed in Definition 4.12. That is, it is not clear if, say, the first non-trivial poset used of the form P(S, G 0 ) is equivalent to a <κ + -directed closed from the point of view of V [G 0 ] (and we suspect it is not, in general). The key to Lemma 4.20 (and to the analogous, but weaker [13,Claim 3.13]) is the flexibility in having G 0 not be decided yet.
Proof of Lemma 4.20 First, we check κ + -Jensen completeness. Since each iterand is <κ-closed and the iteration uses κ-sized supports, the entire iteration is <κ-closed. So by Corollary 2, to show that P is κ + -Jensen complete, it suffices to show that whenever (H(θ ), ∈, P) with |W | = κ and W ∩ κ + ∈ κ + , and • g ⊆ W ∩ P is a (W , P)-generic filter, then g has a lower bound in P. So fix such a filter g for the remainder of the proof. Given α ∈ W ∩ [0, ε], define g α as in Definition 4.15. Set δ = W ∩ κ + . We consider two cases: g 0 , and consider the function p( f 0 , g) from Definition 4.15. We claim that p( f 0 , g) is flat condition and lies below all members of g.
Let p( f 0 , g) be the function defined in Definition 4.15. We claim that p( f 0 , g) is a flat condition that lies below every element of g. Assume not. Then, by Lemma 4.16, there is an α ∈ W ∩ [1, ε) such that p( f 0 , g) α is a condition in P α , and that, by part (2(c)ii) of that lemma, there is some condition q ≤ P α p( f 0 , g) α such that q forces that every proper initial segment of ( Ġ 0 )(δ) is an element of W . But the 0th component of p( f 0 , g) α, and hence of q, extends the function f 0 , and therefore In particular, every proper initial segment of t is an element of W , contrary to our choice of t.
This completes the proof of κ + -Jensen completeness. To see that the flat conditions are dense in P, let p 0 be any condition in P. Fix W ≺ (H(θ ), ∈, P, p 0 ) such that |W | = κ ⊆ W and W ∈ IA κ . By Lemma 2.7 and the <κ-closure of P, there exists a (W , P)generic filter g such that p 0 ∈ g. Note that W ∈ IA κ implies that cof(W ∩ κ + ) = κ. This shows that we can repeat the argument from the above Case 2, define f 0 as above and conclude that the function p( f 0 , g) is a flat condition that is below every member of g and therefore also below p 0 . Lemma 4.20 and Corollary 2 now immediately yield the following corollary:

Corollary 3
The poset P is forcing equivalent to a <κ + -directed closed forcing. In particular, it adds no new sets of size κ, and, in the case κ = ω 1 , it preserves all standard forcing axioms, such as MM ++ .
Remember that the order o( p) of a condition in a poset of the form P(S, G 0 ) was defined in part (4) of Definition 4.8.
Our next task is to prove that tails of the iteration behave nicely. But first we need tail versions of Definition 4.15 and Lemma 4.16. Note that in Definition 4.24 below, since α 0 ≥ 1, the entire filter G 0 has already been determined. So unlike Definition 4.15, the candidate for a condition below g will not involve any f 0 .

Definition 4.24
Suppose that α 0 ∈ [1, ε) and G α 0 is P α 0 -generic over V. Working in V[G α 0 ], suppose that W ≺ (H(θ )[G α 0 ], ∈, P/G α 0 ) with |W | = κ ⊆ W , and g ⊆ W ∩ P/G α 0 is a (W , P/G α 0 )-generic filter. For each α ∈ W ∩ [α 0 , ε), define P α /G α 0namesċ g,α ,ḣ g,α andẊ g,α analogously to Definition 4.15, and define a function p(g) with domain W ∩ [α 0 , ε) by setting We now also have a tail variant of Lemma 4.16: and suppose W and g are as in Definition 4.24. Given α ∈ [α 0 , ε], set Then one of the following statements holds: (1) p(g) is a flat condition in P/G α 0 that extends every element of g. (2) There is an α ∈ W ∩ [α 0 , ε) such that the following statements hold: (a) p(g) α is a flat condition in P α /G α 0 that is stronger than every element of g α . (b) If τ ∈ W is a P α /G α 0 -name for a function from κ to the ordinals, then (c) There is a condition q in P α /G α 0 below p(g) α with the property that the following statements hold true in Proof The proof is almost identical to the proof of Lemma 4.16, except we work in V[G α 0 ] instead of V. We leave the details to the reader.

Lemma 4.26 If α < ε and G is P α -generic over V, then the tail of the iteration
Proof Let q ξ | ξ < μ be a descending sequence with μ < κ in P/G in V[G]. Since P/G ⊆ P and Lemma 4.20 shows that P α is <κ-closed in V, this sequence is an element of V. LetĠ denote the canonical P α -name for the generic filter in V. Fix a condition p in G such that p P α " Ever y condition inĠ is compatible withq ξ inP " holds in V for all ξ < μ. Work in V. Given ξ < μ, a standard density argument now shows that p P α "q ξ α ∈Ġ " and the separativity of P α allows us to conclude that p ≤ P α q α holds. Fix a condition r below p in P α and set for all ξ < μ. Then r ξ | ξ < μ is a descending sequence of conditions in P, and, by the proof of Lemma 4.20, this sequence has a lower bound r μ in P. Then r μ α ≤ P α r and r μ ≤ P q ξ for all ξ < μ. By genericity, we can now find a condition q in P with the property that q α ∈ G and q ≤ P q ξ for all ξ < μ. But then we can conclude that q is a condition in P/G in V[G] with q ≤ P/G q ξ for all ξ < μ.
The proof of the following lemma is similar to the proof of [13, Claim 3.14], but there are some subtle differences since we do not assume that κ <κ = κ. Roughly, we replace their use of κ-properness with IA κ -properness (Definition 2.8) and verify that the argument still goes through.

Lemma 4.27
If α 0 < ε and G is P α 0 -generic over V, then the tail of the iteration P/G is IA κ -totally proper in V[G].
Proof For α 0 = 0, the statement of the lemma follows immediately from Lemmas 2.7 and 4.20. Therefore, we from now on assume that 1 ≤ α 0 < ε.
Let G α 0 be P α 0 -generic over V and work in V[G α 0 ]. Let θ be a sufficiently large regular cardinal, let be a well-ordering of with W ∈ IA κ , and let p 0 be a condition in W ∩ (P/G α 0 ). In the following, we will find a (W , P/G α 0 )-total master condition below p 0 . Define which is well-defined because W ∈ IA κ implies that cof(W ∩ κ + ) = κ.

Case 1:
There exists ζ ∈ W ∩ κ + with t ζ / ∈ W . Since Lemma 4.26 implies that P/G α 0 is <κ-closed, we can apply Lemma 2.7 to find a (W , P/G α 0 )-generic filter g that includes p 0 . Let p(g) be the function defined in Definition 4.24 and assume, towards a contradiction, that p(g) is not a condition in P/G α 0 that is stronger than every element of g. Then, by part (2(c)ii) of Lemma 4.25, there is an α ∈ W ∩ [α 0 , ε) and some condition in P α /G α 0 below p(g) α forcing that every proper initial segment of ( G 0 )(W ∩ κ + ) is an element of W . But this implies that every proper initial segment of t is an element of W , contrary to our case. This allows us to conclude that p(g) is a (W , P/G α 0 )-total master condition below p 0 .
listing all open dense subsets of P/G α 0 that are elements of W , and such that every proper initial segment of D is an element of W . Recursively define a descending sequence p = p ξ | ξ < κ of conditions in P/G α 0 as follows: • Given ξ < κ, let p * ξ be the -least flat condition in P/G α 0 below p ξ that is an element of D ξ . Such a condition exists by Lemma 4.20 and the open density of D ξ . By Lemma 4.23, there exists β < κ + with for all 1 ≤ α ∈ sprt( p * ξ ). Given α ∈ [α 0 , ε), letḟ α andẎ α denote the canonical P α /G α 0 -names with the property that Note that, by the choice of β, for each α ∈ sprt( p * ξ ), the condition p * ξ α forces that β is larger than all domains of elements of the range ofḟ α , and larger than all elements in the domain ofẎ α . In particular, if α 0 ≤ α ∈ sprt( p * ξ ) and G is ( is a condition inQ G α below p * ξ (α) G . 19 This shows that there is a condition p ξ +1 below p * ξ in P with sprt( p ξ +1 ) = sprt( p * ξ ) and the property that whenever then p ξ +1 (α) G is equal to the condition in (10). Now, assume that p ξ is an element of W . Then p * ξ is obviously definable in (H(θ )[G α 0 ], ∈, P, G α 0 , ) using the parameters p ξ and D ξ , which are both contained in W . Since p * ξ ∈ W , then β can also be taken to be an element of W . Finally, the condition p ξ +1 is definable from p * ξ , β, and t (β + 1), all of which are elements of W because of the case we are in. These arguments show that p ξ ∈ W implies that p ξ +1 ∈ W .
• If ξ < κ is a limit ordinal, then we define p ξ be the -least lower bound of the sequence p ζ | ζ < ξ in P. 20 Note that every proper initial segment of p is an element of W , because W contains all proper initial segments of t and each proper initial segment of p is definable in the structure (H(θ )[G α 0 ], ∈, , P) using the parameter p 0 and some sufficiently long 21 proper initial segment of t. Hence, not only is each p ξ +1 an element of D ξ , but is in fact an element of D ξ ∩ W . In particular, the set { p ξ | ξ < κ} generates a (W , P/G α 0 )generic filter. Let g denote this filter, and let p(g) be the function defined in Definition 4.24. Now, assume, towards a contradiction, that p(g) is not a condition below every member of g. Then by part (2(c)iii) of Lemma 4.25, there is an α ∈ W ∩ [α 0 , ε) and a condition q in P α /G α 0 below p(g) α with the property that whenever G is (P α /G α 0 )-generic over V[G α 0 ] with q ∈ G, then no proper initial segment of t = ( G 0 )(W ∩κ + ) is an element of ran(Ẋ G g,α ). Since α ∈ W and the set { p ξ | ξ < κ} generates the (W , P/G α 0 )-generic filter g, we can find ξ α < κ with the property that α ∈ sprt( p ξ α +1 ).
Since p ξ α +1 ∈ g, we know that p ξ α +1 (α) G ∈ċ G g,α and henceẊ G g,α extends the right coordinate of p ξ α +1 (α) G . By construction, the range of the right coordinate of p ξ α +1 (α) G contains a proper initial segment of t, contradicting the properties of α and q.
Again, we can conclude that p(g) is a (W , P/G α 0 )-total master condition below the condition p 0 . Corollary 4 Let G be P-generic over V, let α ∈ (0, ε), let G α be the filter on P α induced by G, and let S =Ṡ G α α . Then . Proof First, if S = ∅, then the conclusion of the lemma holds trivially, because Corol- In the other case, we know that S κ + κ \ S contains a stationary set in Our proof of this lemma is similar to the proof of [13,Claim 3.15], but we must make the following changes: • Whereas the proof of [13,Claim 3.15] makes use of <κ-closed elementary submodels of size κ (whose existence requires the assumption κ <κ = κ), we instead use elementary submodels in IA κ . • We use Corollary 4 to ensure that the complement of eachṠ α is stationary in the final model (this is used to get the right analogue of statement (8) on page 1691 of [13]).

Proof of Lemma 4.28
Letḃ be a P-name for a function from κ + to κ + . Assume, towards a contradiction, that there is a condition p in P that forcesḃ to be a cofinal branch through T (Ġ). Fix W ≺ (H(θ ), ∈, P,ḃ, p) with W ∈ IA κ . Set δ W = W ∩ κ + . By the <κ-closure of P and Lemma 2.7, there exists a (W , P)-generic filter g containing p.
By the (W , P)-genericity of g, the <κ + -distributivity of P, and the fact thatḃ ∈ W , it follows that for every γ < δ W , some condition p γ in g decides the value ofḃ γ . Define Then t is a function from δ W to δ W . Moreover, by the (W , P)-genericity of g and the fact thatḃ is forced to be a branch throughṪ (Ġ 0 ), we know that Hence, if we let g 0 denote the 0-th component of the (W , P)-generic filter g, then we have t γ = ( g 0 )(γ ) for all γ < δ W . It follows that is a condition in Q 0 .
Let p( f 0 , g) be the function defined in Definition 4.15. Then p( f 0 , g) is not a condition in P that extends every element of g, because otherwise it would force thaṫ b δ W = t = ( Ġ 0 )(δ W ) and hence it would also force thatḃ δ W / ∈Ṫ (Ġ 0 ), contradicting our assumptions on p.
In this situation, Lemma 4.16 yields an α ∈ W ∩ [1, ε) such that p( f 0 , g) α is a condition in P α below every member of g α and there is a condition q below p( f 0 , g) α in P α with the property that whenever G is P α -generic over V with q ∈ G, then every proper initial segment of ( G 0 )(δ W ) is an element of W , and is anṠ G α -node. Since the 0-th coordinate of q extends f 0 , we know that Let H W denote the transitive collapse of W , and let σ : Since q extends every element of the (W , P α )-generic filter g α , it follows that G ∩ W = g α , g α isP α -generic over H W , and the map σ can be lifted to an elementarŷ by settingσ (τḡ α ) = (σ (τ )) G for allP α -namesτ in H W . Moreover, we know thatḡ isP-generic over H W and the function  Proof of Claim 4.29 Fix γ < δ W , and set s = t γ . By earlier remarks, we know that s ∈ W and, by the definition of t in (11), there is p ∈ g ⊆ W with p P "ḃ γ =š ". We now know that p,ḃ, γ , and s are elements of W = ran(σ ), and since crit (σ ) = δ W , it follows that σ fixes γ and s. The elementarity of σ now implies that holds in H W . Moreover, since p ∈ g, we have σ −1 ( p) ∈ḡ and henceb γ = s holds in H W [ḡ].
Then b * =b maps from to , and, by the <δ W -distributivity ofP over H W , we know that b * is an element of H W . Moreover, since crit σ = δ W , we havê σ (b * ) = b * . In addition, since = dom(b * ) ∈ (S δ W κ \S) H W [ḡ α ] and is closed under b * , we can conclude that H W [ḡ α ] believes that b * is not aS-node. 22 Then the elementarity ofσ : . By Claim 4.29, we now have b * = t . So t is not aṠ G α -node, contradicting our earlier arguments.
We are now ready to complete the proof of the main technical result of this paper.
Proof of Theorem 4.2 Let κ be an infinite regular cardinal and let P = P ε be the poset constructed in Definition 4.12. Then Lemma 4.20 shows that part (1) of the theorem holds. Next, we prove part (2) of the theorem. We will prove that the poset P is (2 κ ) +stationarily layered (see [3,Definition 29]) in V, which, by [3,Lemma 4], implies that P is (2 κ ) + -Knaster. A poset R is λ-stationarily layered if for some sufficiently large regular cardinal θ , there are stationarily-many M ∈ ℘ * λ (H(θ )) such that M ∩ R is a regular suborder of R. Equivalently, we can demand that every condition p in R has a reduction into M ∩ R, i.e. there exists q ∈ M ∩ R such that all extensions of q in M ∩ R are compatible with p in P.
For all sufficiently large regular cardinals θ , the set We prove that R witnesses the (2 κ ) + -stationary layeredness of P. Fix M ∈ R and a condition p in P. By the density of flat conditions in P, we we may assume that there exists a sequence x α | α ∈ sprt( p) that witnesses that p is flat.
Furthermore, we may assume that holds for all α ∈ sprt( p), because redefining p in this way results in a condition equivalent to p. Set s = M ∩ sprt( p) and define q = p s.
both p and r in P by checking inductively that ( p ∧ r ) β is below both p β and p β for all β ≤ ε. Suppose this statement holds at all α < β ≤ ε. Clearly, if β is a limit ordinal, then it holds at β as well. Hence, we may assume that β = α + 1 and that ( p ∧ r ) α lies below both p α and r α. If α is not in the support of either p or r , then the above statement trivially holds at β as well. Hence, we have to consider the following two cases: Case 1: α ∈ sprt(r ). By definition of the condition p ∧ r , we have ( p ∧ r )(α) = r (α) and r ≤ P q implies that r α P α " r (α) ≤Q α q(α) ". Since ( p ∧ r ) α ≤ P α r α by our induction hypothesis, this shows that Finally, we prove part (3b) of the theorem. Hence, assume that κ <κ ≤ κ + holds in V and fix an enumeration z = z ξ | ξ < κ + of all elements of [κ + ] <κ in V. By Lemma 2.10, the set M of all γ ∈ S κ + κ that are approachable with respect to z is a maximal element of I [κ + ] ∩ ℘ (S κ + κ ) mod N S in V. Since P is <κ + -distributive and therefore z still enumerates all of [κ + ] <κ in V[G], it follows that M is still the set of all γ ∈ S κ + κ that are approachable with respect to z in V [G], and hence M is still a maximal element of

Applications
We now apply Theorem 4.2 to prove the results presented in the introduction of the paper.

Corollary 5
Let κ be an infinite regular cardinal satisfying κ <κ ≤ κ + , let P be the partial order given by Theorem 4

.2 and let M be a maximum element of I
Proof Work in V[G] and let T be the subtree of <κ + κ + given by Theorem 4.2. Then T ⊆ <κ + κ + ∈ H((2 κ ) + ). Define S to be the collection of all subsets A of M such that either there exists a closed unbounded subset C of κ + with C ∩ M ⊆ A or there exists an order-preserving function from the tree T (S κ + κ \ A) into the tree T . Then the set S is definable by a 1 -formula with parameters M, T and <κ + κ + .

Claim 5.1
The set S is equal to the collection of all subsets of M that are stationary in κ + .
Proof of Claim 5.1 First, let A ⊆ M be stationary in κ + with the property that there is no club C in κ + with C ∩ M ⊆ A. Since M is stationary in κ + , this shows that A is bistationary in S κ + κ , M \ A is stationary, and hence Theorem 4.2 yields an orderpreserving function from T (S κ + κ \ A) into T that witnesses that A is contained in S. This argument shows that S contains all stationary subsets of M. Now, assume, towards a contradiction, that there is a non-stationary subset A of κ + that is contained in S. Then there is an order-preserving embedding of T (S κ + κ \ A) into T and a closed unbounded subset C of κ + with A ∩ C = ∅. But then C ∩ S κ + κ is a κ-club that is a subset of S κ + κ \ A and, by earlier remarks, the tree T (S κ + κ \ A) contains a cofinal branch. But then the tree T also contains a cofinal branch, a contradiction.
By the above claim, the set N S M = ℘ (M) \ S is definable by a 1 -formula with parameters in H((2 κ ) + ).
In particular, the above corollary directly shows how the definability results of [13] and [23] can be derived from Theorem 4.2.

Corollary 6
Let κ be an infinite regular cardinal satisfying κ <κ = κ and let P be the poset given by Theorem 4.2. If G is P-generic over V, then N S S κ + κ is 1 (H((2 κ ) Proof By Lemma 2.10, if κ <κ = κ holds in V, then S κ + κ is a maximum element of I [κ + ] ∩ ℘ (S κ + κ ) mod N S. Since forcing with P does not change cofinalities below κ + , the desired conclusion directly follows from Corollary 5.
The following lemma establishes a connection between principles of stationary reflection and the 1 -definability of restrictions of the non-stationary ideals that will be crucial for proofs of our main results. Proof Let S denote the collection of all subsets A of S with the property that there exists E ∈ E such that A ∩ α is stationary in α for all α ∈ E. By our assumptions on E, the set S is definable by a 1 -formula with parameters p, S and H(δ).
If A ⊆ S is stationary in δ, then our assumptions on E ensure that A is contained in S.
In the other direction, if E ∈ E witnesses that A is an element of S and C is closed unbounded in δ, then there is α ∈ E ∩ Lim(C) with A ∩ α stationary in α and hence ∅ = A ∩ C ∩ α ⊆ A ∩ C. Together, this shows that S is equal to the collection of all subsets of S that are stationary in δ and hence N S S = ℘ (δ) \ S is definable by a 1 -formula with parameters p, S and H(δ).
The above lemma directly shows that strong forms of stationary reflection cause restrictions of non-stationary ideals to be 1 -definable.

Corollary 7
Let δ be an uncountable regular cardinal, let E be a stationary subset of S δ >ω and let S be a stationary subset of δ such that every stationary subset of S reflects almost everywhere in E (i.e. for every stationary subset A of S, there is a closed unbounded subset C of δ with the property that A reflects at every element of C ∩ E). Then the set N S S is definable by a 1 -formula with parameters E, S and H(δ).
Proof If we define E = {C ∩ E | C club in δ}, then E is definable by a 1 -formula with parameter E and this shows that the sets E and S satisfy the assumptions of Lemma 5.2.
Note that a classical result of Magidor in [22] shows that, starting with a weakly compact cardinal, it is possible to construct a model of set theory in which every stationary subset of S 2 0 reflects almost everywhere in S 2 1 . The above corollary shows that the set N S S 2 0 is 1 (H(ω 3 ))-definable in Magidor's model. The next theorem will be used to derive Theorem 1.2.
Then there exists a <ω 2 -directed closed, cardinal-preserving poset P with the property that the following statements hold in V[G] whenever G is P-generic over V: (1) 2 ω 2 = θ . (2) If for every stationary subset A of S 2 0 , there is a stationary subset R of IA ω 1 such that W ≺ H (ω 3 ) and A reflects at W ∩ ω 2 for all W ∈ R, then the set N S S 2 0 is 1 (H(ω 3 ))-definable.
Proof Let G be Add(ω 2 , θ)-generic over V. Since 2 ω 1 = ω 2 holds in V, we know that Add(ω 2 , θ) satisfies the ω 3 -chain condition in V and hence all cofinalities are preserved in V [G]. Work in V[G]. Then our assumptions ensure that 2 ω 1 = ω 2 and 2 ω 2 = θ = θ ω 2 . Let P be the poset given by Theorem 4.2 for κ = ω 1 and ε = θ , and let M be a maximum element of I [ω 2 ] ∩ ℘ (S ω 2 ω 1 ) mod N S, which exists due to the assumption that 2 ω ≤ ω 2 (see Lemma 2.10). Then Lemma 3.6 and part (1) of Theorem 4.2 show that P is forcing equivalent to a <ω 2 -directed closed poset. Moreover, since 2 ω 1 = ω 2 holds, part (2) of Theorem 4.2 shows that P satisfies the ω 3 -chain condition. Finally, Lemma 4.20 shows that P has a dense subset of cardinality θ . Now, let H be P-generic over V[G] and work in V[G, H ]. By the above observations, we then have 2 ω 2 = θ . In addition, part (3b) of Theorem 4.2 shows that M is the maximum element of I [ω 2 ] ∩ ℘ (S ω 2 ω 1 ) mod N S. Moreover, Corollary 5 shows that N S M is 1 (H(ω 3 ))-definable. In the following, assume that for every stationary subset A of S 2 0 , there is a stationary subset R of IA ω 1 such that W ≺ H (ω 3 ) and A reflects at W ∩ ω 2 for all W ∈ R. Set E = ℘ (M) \ N S ω 2 . Then E is definable by a 1 -formula with parameters in H(ω 3 ).

Claim 5.4 For every stationary subset A of S 2 0 , there is an element E of E with the property that A reflects at every element of E.
Proof of the Claim By our assumption, there is a stationary subset R of IA ω 1 such that W ≺ H (ω 3 ) and A reflects at W ∩ ω 2 for every W ∈ R. If we now define E 0 = {W ∩ ω 2 | W ∈ R}, then E 0 is a stationary subset of S ω 2 ω 1 . Moreover, since 2 ω ≤ ω 2 , each W ∈ R has (as an element) an enumeration z = z ξ | ξ < ω 2 of [ω 2 ] ω and therefore the internal approachability of W and the fact that z ∈ W imply that W ∩ ω 2 is approachable with respect to z. Hence, the set E 0 is stationary and an element of I [ω 2 ]. Since M is the largest such element mod N S, we have in particular that E = E 0 ∩ M is a stationary subset of M.
Using Lemma 5.2, we can now conclude that N S S 2 0 is definable by a 1 -formula with parameters in H(ω 3 ).
Let θ be a cardinal with θ ω 2 = θ . Since PFA implies that 2 ω = 2 ω 1 = ω 2 holds (see [15,Theorem 16.20 & 31.23]), our assumption allows us to apply Theorem 5.3 to obtain a <ω 2 -directed closed poset with the properties listed in the conclusion of theorem. Let G be P-generic over V. Then [4,Theorem 4.7] ensures that FA holds in V[G]. Since FA holds in V[G], there exists a stationary subset R of IA ω 1 with the property that for all W ∈ R, we have W ≺ H (ω 3 ) and A reflects at W ∩ ω 2 . 23 Then Theorem 5.3 allows us to conclude that N S S 2 0 is 1 (H(ω 3 ))-definable in V[G].
The next theorem will be used to derive Theorem 1.3.
(2) If every stationary subset of S 2 0 reflects to a point in S 2 1 , then the set N S ω 2 is 1 (H(ω 3 ))-definable.
Proof Let G be Add(ω 2 , θ)-generic over V, let P be the poset produced by an application of Theorem 4. Then every stationary subset of S 2 0 reflects to stationary-many points in S 2 1 and we can apply Lemma 5.2 with S 2 0 and ℘ (S 2 1 ) \ N S ω 2 to show that N S S 2 0 is 1 (H(ω 3 ))-definable. Since it is easy to see that N S ω 2 = {A ⊆ ω 2 | A ∩ S 2 0 ∈ N S S 2 0 and A ∩ S 2 1 ∈ N S S 2 1 }, these computations allow us to conclude that N S ω 2 is 1 -definable.
Proof of Theorem 1.3 Assume that 2 ω = ω 1 , 2 ω 1 = ω 2 and either FA + (σ -closed) or SCFA holds. Let θ be a cardinal with θ ω 2 = θ , let P be the poset produced by an application of Theorem 5.5 and let G be P-generic over V. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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